… is the pattern of survival and reproduction.
“pattern” means organized by:
- Age
- Size
- Stage (larva / pupa / adult)
Today mainly:
- Gotelli - Chapter 3
March 7, 2022
… is the pattern of survival and reproduction.
* - Fecundity is number of offspring per individual over some unit of time.
| . | Larva | Sophomore | Emeritus |
|---|---|---|---|
| . | |||
| Survival | 0.5 | 1 | 0 |
| Fecundity | 0 | 1.5 | 0.5 |
See numerical experiment: https://egurarie.shinyapps.io/AgeStructuredGrowth/
| Stage | Survival | Fecundity |
|---|---|---|
| 1. larvae | 0.5 | 0 |
| 2. sophomore | 1 | 2 |
| 3. emeritus | 0 | .5 |
This time, population grows. And the stable age distribution is a bit different.
A “schedule” of all births and deaths in our population
If you know When and With what probability individuals die and when and how many offspring they have, you can compute:
Often - beyond a description - the real question is:
| Age class \((x)\) | Number at Start \((S_x)\) | Cumulative Prob. of survival \((l_x = S_x/S_1)\) | births/ind \((b_x)\) | Surv. prob \((g_x = l_x - l_{x+1})\) | Reproductive rate: \((l\times b)\) | \(g = lbx\) |
|---|---|---|---|---|---|---|
| 1. larva | 16 | 1.0 | 0 | 0.5 | 0 | 0 |
| 2. sophomore | 8 | 0.5 | 1.5 | 1 | 0.75 | 1.5 |
| 3. emeritus | 8 | 0.5 | .5 | 0 | 0.25 | .75 |
| 4. beyond | 0 | 0 | – | – | – | – |
| totals: | \(\Sigma S_i = 32\) | \(\Sigma D_i = S_1 = 16\) | \(R_0 = \sum lb = 1\) | \(G = {\sum(lbx) \over \sum(lb)} = 2.25\) |
| \(x\) | \(S_x\) | \(l_x\) | \(b_x\) | \((l\times b)\) | \(g = lbx\) |
|---|---|---|---|---|---|
| 1. larva | 16 | 1.0 | 0 | 0.5 | 0 |
| 2. sophomore | 8 | 0.5 | 1.5 | 1 | 0.75 |
| 3. emeritus | 8 | 0.5 | .5 | 0 | 0.25 |
| 4. beyond | 0 | 0 | – | – | – |
| totals: | \(\Sigma S_i = 32\) | \(R_0 = 1\) |
Growth rate is approximately: \(r \approx \log(R_0)/G\) but that doesn’t work for our example at all
Growth rate exactly solves: \[1 = \sum_{x=1}^n e^{-rx} l(x) b(x)\] Generally this is hard.
In our example:
\[1 = e^{-r} 0 + e^{-2r} {3 \over 4} + e^{-3r} {1\over 4}\]
eventually leads to: \(r = 0\).
Another important metric: Number of offspring YET to be born.
\[v(x) = {e^{rx}\over l(x)} \sum_{y = x+1}^k e^{-ry} \, l(y)b(y)\]
TYPE I: high survivorship for juveniles; most mortality late in life
TYPE II: survivorship (or mortality) is relatively constant throughout life
TYPE III: low survivorship for juveniles; survivorship high once older ages are reached
| Stage | Survival | Fecundity |
|---|---|---|
| 1. larvae | 0.5 | 0 |
| 2. sophomore | 1 | 2 |
| 3. emeritus | 0 | .5 |
Jargon:
| Stage | Survival | Fecundity |
|---|---|---|
| 1. smolt | 0.05 | 0 |
| 2. ocean | 0.9 | 0 |
| 3. return | 0 | 21 |
Given by:
\[c(x) = {e^{-rx} l(x) \over \sum_{x = 0}^k e^{-rx} l(x)}\]
For M. academicus, pretty easy (since \(r = 0\)), just the \(l(x)\) ratios: 1/2, 1/4, 1/4.
For O. gorbusha, \(r\) is also just about 0 … but is there really a “stable distribution”?
(revisit shiny app: https://egurarie.shinyapps.io/AgeStructuredGrowth/ and see how growth rate affects distribution).
Jargon: if \(r = 0\), we refer to the distribution as “stationary”.
are the same as exponential growth:
The really interesting ecological questions ask what happens when conditions change
Through time: Cohort analysis - follow a cohort (or several cohorts) through time
Snapshot: Age distribution - almost always something weird turns up!
Mark-racapture, observational analysis of survival and reproduction
If you have N-1 variables you can often infer the missing one!
… next week we learn ONE EASY TRICK to get both the intrinsic growth rate and stable distribution and reproductive value from a life table using the: